Question

How many prime numbers p, p^2+ 3p-1 is also a prime number?

Answer 1

Answer:

2+3+1+7=1357$

I hope this is help you

Answer 2

p² + 3p - 1 is prime for p = 17, i.e. one prime number.

Given: prime number p

To find: prime number p where p² + 3p - 1 is a prime number

Solution:

We know that,

Every prime number greater than 3 is of the form 6k ± 1 (k∈N)

1. p = 6k + 1 (say)

⇒ p² + 3p - 1 = (6k + 1)² + 3(6k + 1) - 1

⇒ p² + 3p - 1 = 36k² + 1 + 12k + 18k + 3 - 1

⇒ p² + 3p - 1 = 36k² + 30k + 3

⇒ p² + 3p - 1 = 3(12k² + 10k + 1) = 3q (say)

⇒ p² + 3p - 1 is divisible by 3, hence not a prime number

2. p = 6k - 1 (say)

⇒ p² + 3p - 1 = (6k - 1)² + 3(6k - 1) - 1

⇒ p² + 3p - 1 = 36k² - 12k + 1 + 18k - 3 - 1

⇒ p² + 3p - 1 = 36k² + 6k - 3

⇒ p² + 3p - 1 = 3(12k² + 2k - 1) = 3q (say)

⇒ p² + 3p - 1 is divisible by 3, hence not a prime number

∴ For no prime number p > 3, p² + 3p - 1 is prime

Now, we have two more prime numbers left → 2 and 3

If p = 2,

⇒ p² + 3p - 1 = 4 + 6 - 1 = 9, which is not prime.

If p = 3,

⇒ p² + 3p - 1 = 9 + 9 - 1 = 17, which is prime

⇒ p² + 3p - 1 is prime for p = 17, i.e. one prime number.

SPJ2